We study the bounded endomorphisms of $\ell_{N}^2(G)=\ell^2(G)\times \dots \times\ell^2(G)$ that commute with translations, where $G$ is a discrete abelian group. It is shown that they form a C*-algebra isomorphic to the C*-algebra of $N\times N$ matrices with entries in $L^\infty(\widehat{G})$, where $\widehat{G}$ is the dual space of $G$. Characterizations of when these endomorphisms are invertible, and expressions for their norms and for the norms of their inverses, are given. These results allow us to study Riesz systems that arise from the action of $ G $ on a finite set of elements of a Hilbert space.

中文翻译：

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由阿贝尔群的动作生成的离散卷积算子和 Riesz 系统

我们研究了 $\ell_{N}^2(G)=\ell^2(G)\times \dots \times\ell^2(G)$ 的有界自同态，它与翻译相通，其中 $G$ 是离散阿贝尔群。结果表明，它们形成了与 $N\times N$ 矩阵的 C*-代数同构的 C*-代数，其中条目位于 $L^\infty(\widehat{G})$，其中 $\widehat{G} $ 是 $G$ 的对偶空间。给出了这些自同态何时可逆的特征，以及它们的范数和它们的逆范数的表达式。这些结果使我们能够研究由 $ G $ 对希尔伯特空间的有限元素集的作用而产生的 Riesz 系统。